THE LEARNING LIBRARY
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Make the numbers click.
Clear methods. Small discoveries. Examples you can make your own.
Go a little deeper than the answer.
A GOOD PLACE TO START
The part, the whole,
and the “oh, I get it.”
Percentages become easier when you know which number is the whole. Walk through a discount, one small step at a time.
Read the percentage guide ↗FIND YOUR NEXT LIGHTBULB MOMENT
A shelf for every question.
24 guides · 46 focused examples
Find a percentage without losing the whole
Turn a rate into an amount, keep track of the base, and check a discount before using the answer.
Read the guide ↗Measure change from the right starting point
Separate the amount of a change from its percentage, and understand why reversing a comparison changes the answer.
Read the guide ↗Growth and discounts: work in multipliers
Use factors to apply a change and distinguish that task from measuring a change that already happened.
Read the guide ↗Move between percentages and decimals confidently
Understand why converting a percentage means dividing by 100, including small rates and values above 100%.
Read the guide ↗Four fraction operations, one clear method
Choose the correct denominator strategy for adding, subtracting, multiplying and dividing fractions.
Read the guide ↗Simplify a fraction without changing its value
Use common factors, normalize a negative denominator, and recognize when no more cancellation is possible.
Read the guide ↗Different fractions, the same proportion
Create equivalent fractions, test a proposed pair, and use common denominators to compare quantities.
Read the guide ↗Turn a finite decimal into an exact fraction
Use place value to preserve the written decimal exactly, then reduce common factors.
Read the guide ↗Read ratios as relationships, not totals
Simplify positive ratios, preserve units, and turn a part-to-part ratio into shares of a whole.
Read the guide ↗Read an expression before trying to solve it
Identify coefficients, variables and like terms, and distinguish an expression from an equation.
Read the guide ↗Keep both sides balanced in a linear equation
Collect variable terms, undo constants, and recognize equations with one, no or infinitely many solutions.
Read the guide ↗Find x by undoing one operation at a time
Work backward through a linear equation, keep each line readable, and verify your answer by substitution.
Read the guide ↗Use the discriminant to understand quadratic roots
Find coefficients in standard form, classify the roots, and check the signs in the quadratic formula.
Read the guide ↗Measure area with the right dimensions
Choose the correct formula for a rectangle, triangle or circle and keep squared units separate from lengths.
Read the guide ↗Go from lengths to volume
Calculate cubes, cylinders and spheres while checking that every input uses the same length unit.
Read the guide ↗Use the right-triangle relationship correctly
Identify the hypotenuse, find a missing side, and check whether three lengths can form a right triangle.
Read the guide ↗Choose the average that answers your question
Compare mean, median and mode, including outliers, even-sized lists and multiple modes.
Read the guide ↗Build a reliable scientific-calculator routine
Set the angle unit, group expressions clearly, and use an estimate to catch entry errors.
Read the guide ↗Make sense of sine, cosine and tangent
Connect right-triangle ratios to angle inputs and recognize where tangent is undefined.
Read the guide ↗Use inverse trig to find an angle
Distinguish an inverse function from a reciprocal and understand the principal angle returned by a calculator.
Read the guide ↗Read logarithms as questions about powers
Compare common, natural and custom logarithms, and avoid confusing a base with the input number.
Read the guide ↗Follow compound growth one period at a time
Separate the annual rate from the periodic rate and understand the assumptions behind a growth projection.
Read the guide ↗Read a fixed-payment loan estimate
Connect principal, monthly rate and number of payments, then separate the monthly bill from total interest.
Read the guide ↗Separate the loan payment from the housing estimate
See where down payment, property tax and insurance enter a monthly mortgage estimate.
Read the guide ↗1/2, written as a decimal and percent
Work through 1 divided by 2, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗1/3, written as a decimal and percent
Work through 1 divided by 3, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗1/4, written as a decimal and percent
Work through 1 divided by 4, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗1/5, written as a decimal and percent
Work through 1 divided by 5, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗1/8, written as a decimal and percent
Work through 1 divided by 8, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗3/4, written as a decimal and percent
Work through 3 divided by 4, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗3/5, written as a decimal and percent
Work through 3 divided by 5, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗5/6, written as a decimal and percent
Work through 5 divided by 6, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗5/8, written as a decimal and percent
Work through 5 divided by 8, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗6/7, written as a decimal and percent
Work through 6 divided by 7, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗7/10, written as a decimal and percent
Work through 7 divided by 10, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗7/8, written as a decimal and percent
Work through 7 divided by 8, convert the quotient to a percentage, and distinguish an exact value from a rounded display.
Explore the example ↗Build an equivalent-fraction family for 1/3
Generate exact equivalents of 1/3, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 1/7
Generate exact equivalents of 1/7, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 11/13
Generate exact equivalents of 11/13, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 11/17
Generate exact equivalents of 11/17, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 13/17
Generate exact equivalents of 13/17, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 2/3
Generate exact equivalents of 2/3, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 2/9
Generate exact equivalents of 2/9, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 3/4
Generate exact equivalents of 3/4, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 4/7
Generate exact equivalents of 4/7, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 5/7
Generate exact equivalents of 5/7, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 7/9
Generate exact equivalents of 7/9, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Build an equivalent-fraction family for 8/9
Generate exact equivalents of 8/9, choose a denominator deliberately, and verify a proposed match using cross-products.
Explore the example ↗Four percent: small rate, clear place value
Count hundredths before applying a 4% rate.
Explore the example ↗Nine percent as a decimal multiplier
Keep the leading zero when converting 9% to a multiplier.
Explore the example ↗Change a percentage into a decimal, including fractional rates
A percentage can already contain a decimal point; it still needs division by 100.
Explore the example ↗Convert percentages above 100% into decimals
Use a value greater than one when the rate represents more than a whole.
Explore the example ↗Convert a percent to a decimal with a size check
Connect the fraction over 100 to the decimal you enter in a formula.
Explore the example ↗Make a small percentage usable in a formula
Preserve all the zeros when a rate is less than one percent.
Explore the example ↗One hundred percent is one whole
Understand why 100% becomes 1, and how that helps with growth factors.
Explore the example ↗Enter percentage rates correctly in calculators
Distinguish a percent-labeled field from a decimal-rate expression.
Explore the example ↗Turn a discount percent into a price calculation
Find the decimal saving rate and the separate multiplier for the amount remaining.
Explore the example ↗Write a percentage as a decimal and a fraction
Use place value and fraction reduction to describe the same share three ways.
Explore the example ↗Choose an Algebra 2 tool by the operation
Sort equations, products and factoring questions before entering a problem.
Explore the example ↗A student checklist for calculator-ready algebra
Prepare notation, predict the result type and verify an answer without hiding the algebra.
Explore the example ↗An algebra solver starts with a well-defined problem
Decide whether you need evaluation, simplification or a solution set, then choose a matching calculator.
Explore the example ↗Recognize algebraic expression types before combining them
Classify terms and powers, then use addition or distribution without mixing unlike variable parts.
Explore the example ↗Simplify algebraic fractions while keeping restrictions
Factor before canceling and retain denominator exclusions even after a factor disappears.
Explore the example ↗Divide a polynomial by a binomial, one leading term at a time
Follow long division, preserve missing powers and verify the quotient with the remainder.
Explore the example ↗Factor by reversing multiplication
Extract a common factor first, then look for a quadratic product and check the expansion.
Explore the example ↗Handle both sides of a linear equation
Bring variables together and distinguish a genuine solution from a contradiction or identity.
Explore the example ↗Combine polynomial terms without losing a sign
Add, subtract and multiply expanded expressions while tracking coefficients by power.
Explore the example ↗Enter a quadratic with the correct coefficients
Convert an equation to standard form, then interpret real, repeated or complex roots.
Explore the example ↗Find a polynomial remainder by substitution
Use P(a) for division by x − a, and connect a zero remainder with a factor.
Explore the example ↗Isolate x and verify the original equation
Undo operations in reverse order and make substitution the final step, not an optional extra.
Explore the example ↗No matching reads yet.
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